Fall 2016 Schedule

Sep 29

Bruce Driver, UCSD

Title:

The Makeenko-Migdal
equations for the 2d -Yang-Mills measure

Abstract:

We will discuss the Makeenko--Migdal equation (MM equation) which relates
variations of a "Wilson loop functional" (relative to the Euclidean Yang--Mills
measure) in the neighborhood of a simple crossing to the associated Wilson loops
on either side of the crossing. We will begin by introducing the 2d --
Yang-Mills measure and explaining the necessary background in order to
understand the theorem. The goal is to describe the original heuristic argument
of Makeenko and Migdal and then explain how these arguments can be made rigorous
using stochastic calculus.

Critical lattice models are believed to converge to a free field in the scaling
limit, at or above their critical dimension. This has been (partially)
established for Ising and Phi^4 models for d \geq 4. We describe a simple spin
model from uniform spanning forests in $\Z^d$ whose critical dimension is 4 and
prove that the scaling limit is the bi-Laplacian Gaussian field for $d\ge 4$. At
dimension 4, there is a logarithmic correction for the spin-spin correlation and
the bi-Laplacian Gaussian field is a log correlated field. The proof also
improves the known mean field picture of LERW in d=4, by showing that the
renormalized escape probability (and arm events) of 4D LERW converge to some
"continuum escaping probability". Based on joint works with Greg Lawler and Xin
Sun.

Oct 27

Name, University

Title:

Title here.

Abstract:

Abstract here.

Nov 3

Reza Aghajani, UCSD

Title:

Mean-Field Dynamics of
Load-Balancing Networks with General Service Distributions

Abstract:

We introduce a general
framework for studying a class of randomized load balancing models in a system
with a large number of servers that have generally distributed service times and
use a first-come-first serve policy within each queue. Under fairly general
conditions, we use an interacting measure-valued process representation to
obtain hydrodynamics limits for these models, and establish a propagation of
chaos result. Furthermore, we present a set of partial differential equations
(PDEs) whose solution can be used to approximate the transient behavior of such
systems. We prove that these PDEs have a unique solution, use a numerical scheme
to solve them, and demonstrate the efficacy of these approximations using Monte
Carlo simulations. We also illustrate how the PDE can be used to gain insight
into network performance.

Nov 10

Stephen DeSalvo, UCLA

Title:

Poisson approximation of combinatorial assemblies with low rank

Abstract:

We present a general framework for approximating the component structure of
random combinatorial assemblies when both the size $n$ and the number of
components $k$ is specified. The approach is an extension of the usual saddle
point approximation, and we demonstrate near-universal behavior when the rank $r
:= n-k$ is small relative to $n$ (hence the name `low rank’).

In particular, for $\ell = 1, 2, \ldots$, when $r \asymp n^\alpha$, for $\alpha
\in \left(\frac{\ell}{\ell+1}, \frac{\ell+1}{\ell+2}\right)$, the size~$L_1$ of
the largest component converges in probability to $\ell+2$. When $r \sim t\,
n^{\ell/(\ell+1)}$ for any $t>0$ and any positive integer $\ell$, we have
$\P(L_1 \in \{\ell+1, \ell+2\}) \to 1$. We also obtain as a corollary bounds on
the number of such combinatorial assemblies, which in the special case of set
partitions fills in a countable number of gaps in the asymptotic analysis of
Louchard for Stirling numbers of the second kind.

This is joint work with Richard Arratia.

Nov 17

Name, University

Title:

Title here.

Abstract:

Abstract here.

Dec 1

Name, University

Title:

Title here.

Abstract:

Abstract here.

"Winter" 2017 Schedule

Jan 12

Konstantin Tikhomirov, Princeton

Title:

The spectral gap of
dense random regular graphs

Abstract:

Let G be uniformly distributed on the set of all simple d-regular graphs on n
vertices, and assume d is bigger than some (small) power of n. We show that the
second largest eigenvalue of G is of order √d with probability close to one.
Combined with earlier results covering the case of sparse random graphs, this
settles the problem of estimating the magnitude of the second eigenvalue, up to
a multiplicative constant, for all values of n and d, confirming a conjecture of
Van Vu. Joint work with Pierre Youssef.

Jan 19

Name, University

Title:

Title here.

Abstract:

Abstract here.

Jan 26

Name, University

Title:

Title here.

Abstract:

Abstract here.

Feb 2

Name, University

Title:

Title here.

Abstract:

Abstract here.

Feb 9

Masha Gordina, University of Connecticut

Title:

Couplings for hypoelliptic diffusions

Abstract:

Coupling is a way of constructing Markov processes with
prescribed laws on the same probability space. It is known that the rate of
coupling (how fast you can make two processes meet) of elliptic/Riemannian
diffusions is connected to the geometry of the underlying space. In this talk we
consider coupling of hypoelliptic diffusions (diffusions driven by vector fields
satisfying Hoermander's condition). S. Banerjee and W. Kendall constructed
successful Markovian couplings for a large class of hypoelliptic diffusions. We
use a non-Markovian coupling of Brownian motions on the Heisenberg group, and
then use this coupling to prove analytic gradient estimates for harmonic
functions for the sub-Laplacian.

This talk is based on the joint work with Sayan Banerjee and Phanuel Mariano.

Feb 16

Name, University

Title:

Title here.

Abstract:

Abstract here.

Feb 23

Douglas Rizzolo, University of Deleware

Title:

Diffusions on the
space of interval partitions with Poisson-Dirichlet stationary
distributions

Abstract:

We construct a pair of related diffusions on a space of partitions of the unit
interval whose stationary distributions are the complements of the zero sets of
Brownian motion and Brownian bridge respectively. Our methods can be extended to
construct a class of partition-valued diffusions obtained by decorating the
jumps of a spectrally positive Levy process with independent squared Bessel
excursions. The processes of ranked interval lengths of our partition-valued
diffusions are members of a two parameter family of infinitely many neutral
allele diffusion models introduced by Ethier and Kurtz (1981) and Petrov (2009).
Our construction is a step towards describing a diffusion on the space of real
trees, stationary with respect to the law of the Brownian CRT, whose existence
has been conjectured by Aldous. Based on joint work with N. Forman, S. Pal, and
M. Winkel.

Mar 2

Professor Jiangang Ying,
Fudan University

Title:

On symmetric linear diffusions and related problems.

Abstract:

In this talk, a representation of local and regular Dirichlet forms on real
line, which are associated with symmetric linear diffusions, will be given and
based on this, several related problems will be discussed.

Mar 9

Name, University

Title:

Title here.

Abstract:

Abstract here.

Mar 16

Laurent Sallof-Coste, Cornell University

Title:

Convolution powers of complex valued functions

Abstract:

The study of partial sums of iid sequences is tightly
connected to that of iterated convolutions.

In this talk, I will discuss results that
resemble local limit theorems for iterated convolution of complex valued
functions in the case of $\mathbb Z$ and $\mathbb Z^d$. Similarities and
differences with the probability densities will be in the spotlight.

Spring 2017 Schedule

Apr 6

Name, University

Title:

Title here.

Abstract:

Abstract here.

Apr 13

Name, University

Title:

Title here.

Abstract:

Abstract here.

Apr 20

Name, University

Title:

Title here.

Abstract:

Abstract here.

Apr 27

Name, University

Title:

Title here.

Abstract:

Abstract here.

May 4

Name, University

Title:

Title here.

Abstract:

Abstract here.

May 11

Name, University

Title:

Title here.

Abstract:

Abstract here.

May 18

Martin Tassy, UCLA

Title:

Variational principles
for discrete maps

Abstract:

Previous works
have shown that arctic circle phenomenons and limiting behaviors of some
integrable discrete systems can be explained by a variational principle. In this
talk we will present the first results of the same type for a non-integrable
discrete system: graph homomorphisms form Z^d to a regular tree. We will also
explain how the technique used could be applied to other non-integrable models.

May 25

Name, University

Title:

Title here.

Abstract:

Abstract here.

Jun 1

Amber L. Puha,
California State University San Marcos

Title:

Asymptotically Optimal Policies for Many
Server Queues with Reneging

Abstract:

The aim of this work (joint with Amy Ward (USC, Marshall School of Business)) is
to determine fluid asymptotically optimal policies for many server queues with
general reneging distributions. For exponential reneging distributions, it has
been shown that static priority policies are optimal in a variety of settings,
that include generally distributed interarrival and service times. Moreover, in
these cases, the priority ranking is determined by a simple rule known as the
c-mu-theta rule. For non-exponential reneging distributions, the story is more
complex. We study reneging distributions with monotone hazard rates. For
reneging distributions with bounded, nonincreasing hazard rates, we prove that
static priority is not necessarily asymptotically optimal. We identify a new
class of policies, which we are calling Random Buffer Selection and prove that
these are asymptotically optimal in the fluid limit. We further identify a fluid
approximation for the limiting cost as the optimal value of a certain
optimization problem. For reneging distributions with nondecreasing hazard
rates, our work suggests that static priority policies are in fact optimal, but
the rule for determining the priority ranking seems more complex in general. It
is work in progress to prove this.